# 1.1.3 Special matrices

The zero matrix is any m by n matrix which in which all elements are 0

$$
0=
\begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}
$$

The identity matrix is a matrix where all elements are 0, except the elements on the main diagonal, which are equal to 1

$$
I=
\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}
$$

The identity matrix is always square, and plays the role of the number 1 in matrix maths, meaning that $AI=A=IA$

The diagonal matrix has only elements on the diagonal, and 0 elsewhere

$$
D=
\begin{bmatrix}
d_1 & 0 & 0\\
0 & d_2 & 0\\
0 & 0 & d_3
\end{bmatrix}
$$

A banded matrix is similar to a diagonal matrix, but there are elements on multiple parallel diagonals, for example, the following is tridiagonal (three diagonals):

$$
B=
\begin{bmatrix}
d_1 & a_1 & 0\\
b_1 & d_2 & a_2\\
0 & b_2 & d_3
\end{bmatrix}
$$

The upper triangular matrix is a matrix where there are only non-zero elements in the top triangle of the matrix:

$$
U=
\begin{bmatrix}
a & b & c\\
0 & d & e\\
0 & 0 & f
\end{bmatrix}
$$

A lower triangular matrix is the opposite:

$$
L=
\begin{bmatrix}
a & 0 & 0\\
b & c & 0\\
d & e & f
\end{bmatrix}
$$